The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law

Min Zhao; Changzheng Qu

doi:10.3934/cpaa.2020245

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2021, Volume 20, Issue 7&8: 2857-2883. Doi: 10.3934/cpaa.2020245

This issue Previous Article Remarks on global weak solutions to a two-fluid type model Next Article

The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law

Min Zhao and
Changzheng Qu^,

School of Mathematics and Statistics and Center for Nonlinear Studies, Ningbo University, Ningbo 315211, China

^* Corresponding author
^* Corresponding author

Received: May 31, 2020

Revised: June 30, 2020

Early access: September 2020

Published: August 2021

This work is supported by the NSF-China grant-11631007 and grant-11971251

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In this paper, we provide a classification to the general two-component Novikov-type systems with cubic nonlinearities which admit multi-peaked solutions and $ H^1 $-conservation law. Local well-posedness and wave breaking of solutions to the Cauchy problem of a resulting system from the classification are studied. First, we carry out the classification of the general two-component Novikov-type system based on the existence of two peaked solutions and $ H^1 $-conservation law. The resulting systems contain the two-component integrable Novikov-type systems. Next, we discuss the local well-posedness of Cauchy problem to the resulting systems in Sobolev spaces $ H^s({\mathbb R}) $ with $ s>3/2 $, the approach is based on the new invariant properties, certain estimates for transport equations of the system. In addition, blow up and wave-breaking to the Cauchy problem of a system are studied.

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Mathematics Subject Classification: Primary: 37K06; Secondary: 35Q51.

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